## table of contents

realGTcomputational(3) | LAPACK | realGTcomputational(3) |

# NAME¶

realGTcomputational

# SYNOPSIS¶

## Functions¶

subroutine **sgtcon** (NORM, N, DL, D, DU, DU2, IPIV, ANORM,
RCOND, WORK, IWORK, INFO)

**SGTCON** subroutine **sgtrfs** (TRANS, N, NRHS, DL, D, DU, DLF, DF,
DUF, DU2, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)

**SGTRFS** subroutine **sgttrf** (N, DL, D, DU, DU2, IPIV, INFO)

**SGTTRF** subroutine **sgttrs** (TRANS, N, NRHS, DL, D, DU, DU2, IPIV,
B, LDB, INFO)

**SGTTRS** subroutine **sgtts2** (ITRANS, N, NRHS, DL, D, DU, DU2, IPIV,
B, LDB)

**SGTTS2** solves a system of linear equations with a tridiagonal matrix
using the LU factorization computed by sgttrf.

# Detailed Description¶

This is the group of real computational functions for GT matrices

# Function Documentation¶

## subroutine sgtcon (character NORM, integer N, real, dimension( * ) DL, real, dimension( * ) D, real, dimension( * ) DU, real, dimension( * ) DU2, integer, dimension( * ) IPIV, real ANORM, real RCOND, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶

**SGTCON**

**Purpose:**

SGTCON estimates the reciprocal of the condition number of a real

tridiagonal matrix A using the LU factorization as computed by

SGTTRF.

An estimate is obtained for norm(inv(A)), and the reciprocal of the

condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

**Parameters**

*NORM*

NORM is CHARACTER*1

Specifies whether the 1-norm condition number or the

infinity-norm condition number is required:

= '1' or 'O': 1-norm;

= 'I': Infinity-norm.

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*DL*

DL is REAL array, dimension (N-1)

The (n-1) multipliers that define the matrix L from the

LU factorization of A as computed by SGTTRF.

*D*

D is REAL array, dimension (N)

The n diagonal elements of the upper triangular matrix U from

the LU factorization of A.

*DU*

DU is REAL array, dimension (N-1)

The (n-1) elements of the first superdiagonal of U.

*DU2*

DU2 is REAL array, dimension (N-2)

The (n-2) elements of the second superdiagonal of U.

*IPIV*

IPIV is INTEGER array, dimension (N)

The pivot indices; for 1 <= i <= n, row i of the matrix was

interchanged with row IPIV(i). IPIV(i) will always be either

i or i+1; IPIV(i) = i indicates a row interchange was not

required.

*ANORM*

ANORM is REAL

If NORM = '1' or 'O', the 1-norm of the original matrix A.

If NORM = 'I', the infinity-norm of the original matrix A.

*RCOND*

RCOND is REAL

The reciprocal of the condition number of the matrix A,

computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an

estimate of the 1-norm of inv(A) computed in this routine.

*WORK*

WORK is REAL array, dimension (2*N)

*IWORK*

IWORK is INTEGER array, dimension (N)

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine sgtrfs (character TRANS, integer N, integer NRHS, real, dimension( * ) DL, real, dimension( * ) D, real, dimension( * ) DU, real, dimension( * ) DLF, real, dimension( * ) DF, real, dimension( * ) DUF, real, dimension( * ) DU2, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶

**SGTRFS**

**Purpose:**

SGTRFS improves the computed solution to a system of linear

equations when the coefficient matrix is tridiagonal, and provides

error bounds and backward error estimates for the solution.

**Parameters**

*TRANS*

TRANS is CHARACTER*1

Specifies the form of the system of equations:

= 'N': A * X = B (No transpose)

= 'T': A**T * X = B (Transpose)

= 'C': A**H * X = B (Conjugate transpose = Transpose)

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*NRHS*

NRHS is INTEGER

The number of right hand sides, i.e., the number of columns

of the matrix B. NRHS >= 0.

*DL*

DL is REAL array, dimension (N-1)

The (n-1) subdiagonal elements of A.

*D*

D is REAL array, dimension (N)

The diagonal elements of A.

*DU*

DU is REAL array, dimension (N-1)

The (n-1) superdiagonal elements of A.

*DLF*

DLF is REAL array, dimension (N-1)

The (n-1) multipliers that define the matrix L from the

LU factorization of A as computed by SGTTRF.

*DF*

DF is REAL array, dimension (N)

The n diagonal elements of the upper triangular matrix U from

the LU factorization of A.

*DUF*

DUF is REAL array, dimension (N-1)

The (n-1) elements of the first superdiagonal of U.

*DU2*

DU2 is REAL array, dimension (N-2)

The (n-2) elements of the second superdiagonal of U.

*IPIV*

IPIV is INTEGER array, dimension (N)

The pivot indices; for 1 <= i <= n, row i of the matrix was

interchanged with row IPIV(i). IPIV(i) will always be either

i or i+1; IPIV(i) = i indicates a row interchange was not

required.

*B*

B is REAL array, dimension (LDB,NRHS)

The right hand side matrix B.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

*X*

X is REAL array, dimension (LDX,NRHS)

On entry, the solution matrix X, as computed by SGTTRS.

On exit, the improved solution matrix X.

*LDX*

LDX is INTEGER

The leading dimension of the array X. LDX >= max(1,N).

*FERR*

FERR is REAL array, dimension (NRHS)

The estimated forward error bound for each solution vector

X(j) (the j-th column of the solution matrix X).

If XTRUE is the true solution corresponding to X(j), FERR(j)

is an estimated upper bound for the magnitude of the largest

element in (X(j) - XTRUE) divided by the magnitude of the

largest element in X(j). The estimate is as reliable as

the estimate for RCOND, and is almost always a slight

overestimate of the true error.

*BERR*

BERR is REAL array, dimension (NRHS)

The componentwise relative backward error of each solution

vector X(j) (i.e., the smallest relative change in

any element of A or B that makes X(j) an exact solution).

*WORK*

WORK is REAL array, dimension (3*N)

*IWORK*

IWORK is INTEGER array, dimension (N)

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Internal Parameters:**

ITMAX is the maximum number of steps of iterative refinement.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine sgttrf (integer N, real, dimension( * ) DL, real, dimension( * ) D, real, dimension( * ) DU, real, dimension( * ) DU2, integer, dimension( * ) IPIV, integer INFO)¶

**SGTTRF**

**Purpose:**

SGTTRF computes an LU factorization of a real tridiagonal matrix A

using elimination with partial pivoting and row interchanges.

The factorization has the form

A = L * U

where L is a product of permutation and unit lower bidiagonal

matrices and U is upper triangular with nonzeros in only the main

diagonal and first two superdiagonals.

**Parameters**

*N*

N is INTEGER

The order of the matrix A.

*DL*

DL is REAL array, dimension (N-1)

On entry, DL must contain the (n-1) sub-diagonal elements of

A.

On exit, DL is overwritten by the (n-1) multipliers that

define the matrix L from the LU factorization of A.

*D*

D is REAL array, dimension (N)

On entry, D must contain the diagonal elements of A.

On exit, D is overwritten by the n diagonal elements of the

upper triangular matrix U from the LU factorization of A.

*DU*

DU is REAL array, dimension (N-1)

On entry, DU must contain the (n-1) super-diagonal elements

of A.

On exit, DU is overwritten by the (n-1) elements of the first

super-diagonal of U.

*DU2*

DU2 is REAL array, dimension (N-2)

On exit, DU2 is overwritten by the (n-2) elements of the

second super-diagonal of U.

*IPIV*

IPIV is INTEGER array, dimension (N)

The pivot indices; for 1 <= i <= n, row i of the matrix was

interchanged with row IPIV(i). IPIV(i) will always be either

i or i+1; IPIV(i) = i indicates a row interchange was not

required.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -k, the k-th argument had an illegal value

> 0: if INFO = k, U(k,k) is exactly zero. The factorization

has been completed, but the factor U is exactly

singular, and division by zero will occur if it is used

to solve a system of equations.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine sgttrs (character TRANS, integer N, integer NRHS, real, dimension( * ) DL, real, dimension( * ) D, real, dimension( * ) DU, real, dimension( * ) DU2, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, integer INFO)¶

**SGTTRS**

**Purpose:**

SGTTRS solves one of the systems of equations

A*X = B or A**T*X = B,

with a tridiagonal matrix A using the LU factorization computed

by SGTTRF.

**Parameters**

*TRANS*

TRANS is CHARACTER*1

Specifies the form of the system of equations.

= 'N': A * X = B (No transpose)

= 'T': A**T* X = B (Transpose)

= 'C': A**T* X = B (Conjugate transpose = Transpose)

*N*

N is INTEGER

The order of the matrix A.

*NRHS*

NRHS is INTEGER

The number of right hand sides, i.e., the number of columns

of the matrix B. NRHS >= 0.

*DL*

DL is REAL array, dimension (N-1)

The (n-1) multipliers that define the matrix L from the

LU factorization of A.

*D*

D is REAL array, dimension (N)

The n diagonal elements of the upper triangular matrix U from

the LU factorization of A.

*DU*

DU is REAL array, dimension (N-1)

The (n-1) elements of the first super-diagonal of U.

*DU2*

DU2 is REAL array, dimension (N-2)

The (n-2) elements of the second super-diagonal of U.

*IPIV*

IPIV is INTEGER array, dimension (N)

The pivot indices; for 1 <= i <= n, row i of the matrix was

interchanged with row IPIV(i). IPIV(i) will always be either

i or i+1; IPIV(i) = i indicates a row interchange was not

required.

*B*

B is REAL array, dimension (LDB,NRHS)

On entry, the matrix of right hand side vectors B.

On exit, B is overwritten by the solution vectors X.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine sgtts2 (integer ITRANS, integer N, integer NRHS, real, dimension( * ) DL, real, dimension( * ) D, real, dimension( * ) DU, real, dimension( * ) DU2, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB)¶

**SGTTS2** solves a system of linear equations with a
tridiagonal matrix using the LU factorization computed by sgttrf.

**Purpose:**

SGTTS2 solves one of the systems of equations

A*X = B or A**T*X = B,

with a tridiagonal matrix A using the LU factorization computed

by SGTTRF.

**Parameters**

*ITRANS*

ITRANS is INTEGER

Specifies the form of the system of equations.

= 0: A * X = B (No transpose)

= 1: A**T* X = B (Transpose)

= 2: A**T* X = B (Conjugate transpose = Transpose)

*N*

N is INTEGER

The order of the matrix A.

*NRHS*

NRHS is INTEGER

The number of right hand sides, i.e., the number of columns

of the matrix B. NRHS >= 0.

*DL*

DL is REAL array, dimension (N-1)

The (n-1) multipliers that define the matrix L from the

LU factorization of A.

*D*

D is REAL array, dimension (N)

The n diagonal elements of the upper triangular matrix U from

the LU factorization of A.

*DU*

DU is REAL array, dimension (N-1)

The (n-1) elements of the first super-diagonal of U.

*DU2*

DU2 is REAL array, dimension (N-2)

The (n-2) elements of the second super-diagonal of U.

*IPIV*

IPIV is INTEGER array, dimension (N)

The pivot indices; for 1 <= i <= n, row i of the matrix was

interchanged with row IPIV(i). IPIV(i) will always be either

i or i+1; IPIV(i) = i indicates a row interchange was not

required.

*B*

B is REAL array, dimension (LDB,NRHS)

On entry, the matrix of right hand side vectors B.

On exit, B is overwritten by the solution vectors X.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

# Author¶

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